1. Field of the Invention
The present invention relates to a method and system for image reconstruction in fan-or cone-beam X-ray computed tomography and, in particular, to a method and system for reconstructing images using weighting coefficients to weight exposure data.
2. Discussion of the Background
Fan- and cone-beam computed tomography (CT) reconstructs the interior of an object of interest or patient from one-dimensional and two-dimensional projections, respectively, of transmitted x-rays through the object of interest or patient. An x-ray source and an x-ray detector are arranged in a number of different positions so that x-rays transmitted through the object of interest are received at the detector. The detector, either alone or in conjunction with other devices, generates image data for each position of the source and/or detector. The image data is then stored, manipulated, and/or analyzed to reconstruct the interior of the object. In a fan-beam system, the detector forms a linear array of x-ray sensing elements while in a cone-beam system, the detector forms an array of x-ray sensing elements.
The classical path of the x-ray source and detector is along a complete circular orbit, i.e. 360xc2x0, about the object of interest. The source and detector are mechanically joined so as to maintain a constant separation distance and position relative to each other and then revolved around the object.
As shown in FIG. 1, an X-ray source S emits either a cone- or a fan-beam of X-rays toward a detector D. The X-rays emitted by source S are incident upon a three-dimensional object of interest (not shown) such as a calibration phantom, a patient, a test object, or other article of interest. At least a portion of the X-rays generated at point source S pass through or around the object and are received at the detector D. The source S and the detector D are fixed relative to one another and revolve in a substantially circular orbit about an axis A in, for example, a C-arm gantry or ring gantry device. The angular position of the X-ray source S is illustrated here as the angle xcex2 relative to an arbitrary half-line L that terminates at the rotation axis A.
Several disadvantages of complete circular orbits of the source and detector about the object arise due to the nature of the complete orbit itself. Electrical leads must be capable of circumscribing one or more complete revolutions about the object of interest. In medical CT, since the patient must be contained within the orbiting detector and source, access to the patient by medical personnel is hindered. Furthermore, many patients dislike being enclosed within the CT mechanism for the extended times necessary to gather sufficient image data for meaningful reconstruction.
In fan-beam CT, the detector D is a substantially linear array of detector elements typically in arc form on the array source. In cone-beam CT, detector D is an area array of detector elements. Curved line and curved surface arrays of detector elements are also suitable for use as detector element D. In all of these cases, detector element D will have a cross sectional area with a width W in a plane orthogonal to the axis of rotation A. In this particular embodiment, the midpoint of the width of a linear array detector D is substantially positioned at a line N passing through the center of the source S and the axis A.
The angle xcex3 illustrated in FIG. 1 describes the angle of a ray O joining the source S and one element selected from the matrix of elements that constitutes the detector D. In fan-beam CT, the angle xcex3m describes the rays M with the largest (maximum) angle relative to the line N, where the ray M is emitted by the source S and received by the detector D. The physical limit on ray M and hence angle xcex3m can arise due to, for example, the finite length of the detector D (as illustrated), collimation of the source emission (not shown), or the non-omnidirectional emission of X-rays by the source S (also not shown). In FIG. 1 with the midpoint of the cross-sectional area of detector D located at line N, the angle xcex3m on one side of the axis is equal and opposite to angle xcex3m on the other side of the axis. Shifting the detector D relative to line N will change this relationship between the two xcex3m""s and can be accounted for using traditional geometric rules.
FIGS. 2a-c illustrate three example rays Oa, Ob, and Oc over which the same x-ray transmittance is measured at two different angular positions of the source xcex2 relative to an arbitrary half-line L and fan beam angles xcex3. For illustrative purposes, the first angular positions of the source xcex2 is equal to zero in all three examples. In FIG. 2a, ray Oa is the first ray sampled twice, while FIGS. 2b and 2c show respective rays Ob and Oc that are sampled twice at other positions.
In recent years, there has been an attempt to implement fan- and cone-beam CT on gantries that only revolve around a portion of the object or patient during imaging. Such partial orbits are capable of providing complete image data for reconstruction of the interior of an object since many views in a complete circular orbit are redundant, i.e., the image data provide little or no new information. For example, if the object of interest is immobile and the system is ideal (i.e., no noise), switching the location of the source and detector will provide no new information along the ray through the axis even though image data from a second view has been collected.
The advantages of such partial orbits include easier and less expensive mechanical realization, providing access to a patient during medical imaging and enabling supporting mechanisms for the source and detector that do not require complete enclosure of the patient. Also, it allows the use of x-ray imaging and primarily designed for non-CT imaging application to also be used to obtain a CT-image for special needs.
A method for reconstruction of one particular partial orbit, namely an orbit that covers the xe2x80x9cminimal complete data setxe2x80x9d has been described by Dennis Parker (xe2x80x9cOptimal Short Scan Convolution Reconstruction for Fanbeam CT,xe2x80x9d Med. Phys. 9, 254-257, 1982) which is incorporated herein by reference. The xe2x80x9cminimal complete data setxe2x80x9d is the collection of equally-spaced projection image data that can be used in conventional, convolution type, reconstruction methods. The xe2x80x9cminimal complete data setxe2x80x9d spans more than one half of a complete orbit. Namely, it spans 180xc2x0 plus the maximum fan angle 2xcex3m, where the maximum channel angle xcex3m is the largest angle of a ray emitted by the X-ray source that is received at the (substantially two- or three-dimensional) X-ray detector relative to the ray emitted from the source that passes through the axis of rotation of the X-ray source and detector. FIG. 1 schematically illustrates this and other terminology used to describe the current invention.
One disadvantage with the use of such a xe2x80x9cminimal complete data setxe2x80x9d orbit lies in the fact that certain rays are sampled twice as often as other rays. In other words, certain image data is collected twice as often as other image data and are redundant. Illustrative examples are illustrated diagrammatically in FIGS. 2a-c. This can be better illustrated in FIG. 3, where the image data is represented in Radon space. The horizontal axis in FIG. 3 corresponds to the channel angle xcex3, the vertical axis corresponds to the angular position xcex2 of the x-ray source, and, in an actual Radon space representation of a collection of x-ray image data, the grey level of each point in Radon space would correspond to the line integral of the x-ray transmittance along the particular ray defined by the fan angle xcex3 and the angular position of the source xcex2. FIG. 3. indicates the angular positions of the source and the channel angles for rays that are sampled shown by shaded regions during the collection of a xe2x80x9cminimal complete data setxe2x80x9d partial orbit (including those rays illustrated in FIGS. 2a-2c). Such Radon space representations of image data are well-known in the art, and a more complete explanation of these representations can be found in several textbooks including xe2x80x9cImage Reconstruction From Projections: The Fundamentals of Computerized Tomographyxe2x80x9d by Gabor T. Herman, Academic Press, New York, 1980, p. 36-39 and 161-165, the entire contents of which is incorporated herein by reference. In general, the line integrals along the rays p(xcex2,xcex3) and p(xcfx80+xcex2+2xcex3,xe2x88x92xcex3) are equivalent. When the total collection of image data is used to reconstruct the interior of an object, the twice collected image data distorts the appearance of the final image and yields poor quality images.
Various methods and devices for solving this problem with the minimal complete data set have been proposed and implemented. The image data can be rebinned into parallel ray data sets and then analyzed, but this requires further computational effort and time. Naparstek described several alternate methods (IEEE Trans. Nucl. Sci. NS-27, p. 1112 ff., 1980, which is hereby incorporated by reference) that, however, provided inadequate results.
Parker has described a method for solving the problem of oversampling certain ray lines during minimal data reconstruction in fan-beam computed tomography with the divergent beam geometry by introducing weights for the oversampled image data. These weights are required to satisfy Equation (1), namely that
w(xcex2,xcex3)+w(xcfx80+xcex2+2xcex3,xe2x88x92xcex3)=1xe2x80x83xe2x80x83(1)
Parker or Crawford and King (xe2x80x9cComputed Tomography Scanning with Simultaneous Patient Translationxe2x80x9d Med. Phys. 17, 967-982, 1990 and incorporated herein by reference) give explicit formulae for the weights.
Unfortunately, simple and elegant methods and devices for reconstruction using partial orbits intermediate to the minimal complete data set and the complete orbit have yet to be developed.
Accordingly, one object of this invention is to provide a method and system for performing CT imaging that does not require highly precise knowledge of the fan angle.
Another object of this invention is to is to provide a method and system for performing CT imaging that does not require precise synchronization of the gantry rotation and X-ray emission.
A further object of this invention is to is to provide a method and system for image reconstruction using partial orbits where the spanned angle is intermediate to the spanned angle of the minimal complete data set orbit and the spanned angle of a complete orbit.
A still further object of the invention is to provide a method and system that provides a tradeoff between suppressing the signal-to-noise ratio and image resolution.
Yet another object of this invention is to provide a method and system for using partial orbits where the spanned angle is intermediate to the spanned angle of the minimal complete data set orbit and the spanned angle of a complete orbit that requires no additional complexity beyond the Parker method for the minimal complete data set in the case of helical cone beam CT.
A further object of the invention is to use as much data as possible for reconstruction while eliminating or minimizing the extent of data extrapolation in the case of a helical cone beam CT.
A yet still further object of this invention is to is to provide comparable methods and systems for use with complete orbits.
These objects can be realized by an image reconstruction method and system that uses a xe2x80x9cvirtualxe2x80x9d fan angle that is equal to the angular rotation about the axis beyond 180xc2x0 (regardless of the actual fan angle). Alternatively, the xe2x80x9cvirtualxe2x80x9d fan angle is defined as a selected angle that is less than the angular rotation about the axis beyond 180xc2x0, but still larger than the angle for collection of the minimal complete data set. The only constraint on the virtual fan angle in both cases is that it is larger than the physical fan angle of the instrument or, in other words, the exposure path is intermediate to the minimal complete data set and a 360xc2x0 path. The virtual fan angle can be used to calculate weights for the oversampled rays that are used during reconstruction of the interior of objects.
Specifically, a system using such a virtual fan angle will include an X-ray tomograph configured to produce an exposure path of a source about an object of interest that is less than 360xc2x0 but greater than 180xc2x0 plus the fan angle. The virtual fan angle can be determined using either the actual angle spanned by such an orbit or a selected smaller angle still larger than the angle necessary to collect the minimal complete data set, and will be used to determine non-uniform weights for the data collected from rays through the orbit and/or to identify the rays to which such weights will be applied. The weighted data can then be used to reconstruct an image according to any of a number of different reconstruction methods.